A right circular cylinder has a height of 12 and a base radius of r. A sphere has a radius… — SAT

A right circular cylinder has a height of $12$ and a base radius of $r$ . A sphere has a radius of $r$ . If the volume of the cylinder is equal to the volume of the sphere, what is the value of $r$ ?

Answer: 9

Answer

The volume of a cylinder is given by

V = \pi r^2 h

and the volume of a sphere is given by

V = \frac{4}{3}\pi r^3

. Given that the cylinder's height is

12

, its volume is

12\pi r^2

. Setting the volumes equal yields

12\pi r^2 = \frac{4}{3}\pi r^3

. Since

r

is a non-zero radius, we can divide both sides by

\pi r^2

, resulting in

12 = \frac{4}{3}r

. Multiplying both sides by

\frac{3}{4}

gives

r = 9

Step-by-Step Solution

State the standard volume formulas for a right circular cylinder and a sphere.

V_{\text{cylinder}} = \pi r^2 h

and

V_{\text{sphere}} = \frac{4}{3}\pi r^3

These formulas are needed to express the volumes of both solids in terms of

r

Equate the volume of the cylinder to the volume of the sphere and substitute the given height of

12

12\pi r^2 = \frac{4}{3}\pi r^3

The problem states that the volume of the cylinder is equal to the volume of the sphere.

Divide both sides of the equation by

\pi r^2

12 = \frac{4}{3}r

Simplifies the equation to a first-degree equation in terms of

r

Solve for

r

by multiplying both sides of the simplified equation by the reciprocal of

\frac{4}{3}

, which is

\frac{3}{4}

r = 9

Isolates the variable

r

to find the correct value.

Key Concept

Equating the volumes of geometric solids to solve for an unknown dimension.

Estimated Time:1m 30s

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